Events
Department of Mathematics and Statistics
Texas Tech University
 | Monday
Feb. 24
2:00 PM
MATH 014
|
| Real-Algebraic Geometry
Germs
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
|
N/A
N/A
It's safe to say that video games are more popular today than they've ever been before. Likewise, thanks to constantly improving technology making a video game is arguably easier than it's ever been before. These two factors combined mean that the video game market is booming, and almost anyone - from the biggest publisher to a lone guy coding from his parents' basement - can throw their hat in the ring now. Making a game is easy - any idiot can cobble together some Unity store assets and sell it - but making a good game requires more than just coding ability. It requires creativity, effort, and a surprising amount of math. The gameplay is just as important, if not more so, than the story or graphics – and a few badly thought out numbers can throw off the balance of even the most intricate and otherwise fun games. So, in order to help my own games avoid such pitfalls, I’ve been researching the many games that came before to find out what worked and what did not, and where the math behind them led to both elegant, engaging gameplay and frustratingly broken experiences, sometimes within the same game.The subject of the talk are biological and biomedical problems that we model using systems of chemotactic equations with cross-diffusion terms. I will present recent results related to quasi-periodic behavior of biological system arising from chemotactic framework of Keller-Segel.
In the first part of the talk, model representing traveling wave phenomena of bacteria dynamics in the presence of the diffusion in the media source of attraction (e.g. food, drug, etc.) will be presented. We use celebrated Keller-Segel framework using system of partial differential equation with cross-diffusion and reactive terms in case of traveling wave pattern stability. We use closed form type solution of Keller-Segel system when diffusion coefficient for source of the attractor is equal zero. Stability of the traveling band type solution for complete dynamical system with respect to parameters of equation initial and boundary data is investigated in detail. For this purpose Lyapunov type functional is constructed and non-linear Gronwall differential inequality for Energy functional is derived. Obtained result provide explicit estimate for differences between base-line solution with no diffusion in source term and solution of complete realistic system of equation.
Second one is joint research with Angie Piec and Bobby Nettle and dedicated to light driven spatial Algae-Daphinia dynamics as primitive evolutionary model with chemotaxis. First constructed analytically time and spatially dependent solution of system of two equation, which model Algae-Daphinia dynamics and proved using maximum principle machinery that this solution is unique.
Then I proved that this solution is stable depending on the relations between chemotactic and diffusion coefficients, and reactive terms, using Sobolev embedding theorems, and Energy functional.